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# A characterization of right triangles II

Recently the following problem was posted on a Facebook group: Right now we’ll use this problem to obtain another characterization of right triangles, in addition to the ones here.

## Stretched image In any triangle, the ratio of the length of an internal altitude to the “length” of a right bisector is always sandwiched between and .

## Strict inequalities

In general, the ratio under consideration lies strictly between and .

In triangle , let be the foot of the altitude from vertex and the midpoint of . Suppose that the right bisector of intersects side at . If , PROVE that: .

Details here.

In triangle , let be the foot of the altitude from vertex and the midpoint of . Suppose that the right bisector of intersects side at . If , PROVE that: .

Details here.

In triangle , let be the foot of the altitude from vertex and the midpoint of . Suppose that the right bisector of intersects side at . If , PROVE that: .

Details here.

Together we have: .

## Special instance

When the ratio under consideration equals .

Suppose that is a right triangle in which and that is the midpoint of . If the right bisector of intersects side at and , PROVE that: .

Use similar triangles. Or else use the fact that: from example 1. Take so that . Simplify to get .

In triangle , let be the foot of the altitude from vertex and the midpoint of . Suppose that the right bisector of intersects side at . If and , PROVE that is a right triangle with .

Details here.

## Takeaway

Let be a triangle in which is the foot of the internal altitude from vertex , is the midpoint of side , and the right bisector of intersects (or , depending on which is greater) at . Then the following statements are equivalent:

1. is a right triangle in which 2. .

• (Silver ratio) Suppose that is a triangle containing an obtuse angle . Let be the foot of the altitude from and let the right bisector of intersect at .
1. PROVE that , where is the midpoint of .
2. Find a triangle for which , the silver ratio.